Mathy thing I learned today

As the size q of a finite field Fq grows larger, the number of roots of a random polynomial from Fq[x] asymptotically becomes a Poisson distribution with mean value 1. That is, for large finite fields, a random polynomial can be expected to have just one root within the finite field.

So if random matrices give rise to uniformly distributed characteristic polynomials (which I'm not at all sure is the case) then we can expect random matrices over finite fields to have just one eigenvalue, on average.